| L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s + i·8-s − 9-s + 4·11-s + i·12-s − 2i·13-s + 16-s − i·17-s + i·18-s − 4·19-s − 4i·22-s + 24-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s + 0.353i·8-s − 0.333·9-s + 1.20·11-s + 0.288i·12-s − 0.554i·13-s + 0.250·16-s − 0.242i·17-s + 0.235i·18-s − 0.917·19-s − 0.852i·22-s + 0.204·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.491646708\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.491646708\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 17 | \( 1 + iT \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.634357428742950836229566907781, −7.989757574387365598442964700160, −6.95805976449372361837424040819, −6.34909153615327659351796855786, −5.42294632743419596965832574709, −4.42594261847773308528203492736, −3.61090425059723521153176237571, −2.61511717379824294063797066338, −1.65516391802075173741387878117, −0.54509965950568993015144627592,
1.27540490215544479470173831041, 2.74497980197667083150885584168, 3.97416455850201745361862764946, 4.38015416747897394425645690374, 5.32752237855133606322428625067, 6.44662275511855786642609782790, 6.57062747236707499110170805583, 7.75770703475408108362581473403, 8.613504799018316812866139079237, 8.995115335819327596569615436316
